PDE is a very active and vibrant field. It is certainly one of the central themes of modern mathematics, with connections to numerous other branches of mathematics. PDEs are often motivated by applications, and if you are working on PDEs you will likely find yourself interacting with scientists from other disciplines. Often, there is a certain learning curve one has to master to work on PDEs. At first, the area may seem a bit technical and too varied, even messy. However, after covering this curve the subject becomes very rewarding. Once central ideas and connections become clear, technicality and variety turn into depth and richness. Learning some new partial differential equation well becomes like making a new good friend - and the fact that there are many different PDEs becomes a big positive. Another positive aspect of working in PDE is that PDE experts are in high demand in academia. Should you decide to transition to industry, PDE background will also be helpful due to connections to applications.

If you are interested in analysis and PDE or undecided on what research you would like to do, please feel free to email me or stop by so that we can chat. In order that this conversation can have more substance, let me state some principles which I follow when mentoring research of a junior colleague. I will describe some of the directions of my research in more detail after that.

- I think it important that you have some research problems in mind early. Even if you need to learn and read through some papers and books before starting research, having a problem in mind can significantly increase your motivation.
- While it is important to get into research mode early, it is also important to accumulate enough knowledge to start seeing various directions and open problems around your work. This helps you find best directions to work on and build more relationships and collaborations with other mathematicians later.
- Career planning is important. It is best to learn about all tracks available to you: research, teaching or industry. It is good to have some idea about your most likely direction at least a year or two before graduation.
- Everybody is different. What works for me may not work for you and other way around. When I was in graduate school, abstract algebra was hard for me. It is difficult for me to prove something if I can't draw a picture or build a simplified model example to work out first. Fortunately, midway through the first year, I became friends with another student, who was great in algebra. He just somehow knew what to do. But he did not like geometry. He could not fathom how someone can prove something by using pictures. Ever since, we collaborated. Life became wonderfully easy, and we learned a lot from each other. Moral? Talk to others about your math strengths and struggles. How I handle math may not work for you, but I likely have seen enough styles of doing math to give you some advice.
- As should be clear from the previous paragraph, mathematics research benefits greatly from talking to people about mathematics. I usually meet with my students or postdocs at least once a week to discuss math. We have one of the nation's strongest groups in analysis and PDE at Madison by research reputation (US News has ranked us 7th and 10th in the nation in most recent rankings), so there are plenty of high quality people for you to interact with.

Now a bit more about my research. Partial Differential equations is an exceedingly diverse field, with very different subareas. The reason is that many processes in nature seem to be described well by PDE - a realization that probably goes back to Newton. Since nature is so rich, there are many different kinds of partial differential equations, and different tools are needed to analyze them. Calculus of variations, comparison principles for the solutions, Fourier analysis, functional analysis, stochastic calculus, ideas from geometry are just some examples of the tool sets that are used to analyze PDEs. So as someone working in PDEs, I use whatever it takes. There are two ideals that I usually try to keep in mind. I like it when mathematics is rich and beautiful, and I like when it refers to or is at least motivated by real life phenomena - problems connected to physics, biology or economics. It is not always easy to balance these two aspirations! One of my favorite tools is Fourier analysis, because it is so abundant with deep and elegant techniques and ideas.

After being pioneered by Joseph Fourier in early 1900s, Fourier analysis went on to become one of the dominant themes in mathematics and its applications for the next two centuries. It is applied widely to partial differential equations (Fourier transform turns differentiation into multiplication, and that is often beneficial - but applications get a lot richer and subtler than that). It often plays an important role in numerical simulations of PDEs, too. Fourier analysis methods are also used heavily in number theory, ergodic theory and occasionally in almost every other area of mathematics. The methods continue to develop and grow more sophisticated. Out of the four 2010 Fields medalists, for example, Smirnov is a Fourier analyst by training, and Lindenstrauss uses Fourier analysis methods extensively in his work - even though they got their awards for work done in other areas (percolation theory and statistical physics for Smirnov, ergodic theory and number theory for Lindestrauss). The set of ideas and tools that you learn in Fourier analysis just seems to be extremely adaptable and useful in all sorts of problems. Learning these techniques prepares your brain for virtually anything mathematical in a very efficient way.

In any case here are several areas that are currently central in my research.

Mathematical Fluid Mechanics

Mixing and enhancement of diffusion by fluid flow

Mathematical Biology

Quantum mechanics, Schrodinger operators and spectral theory